The general displacement of a simple harmonic oscillator is x = A sin ω t . Let T be its time period. The slope of its potential energy ( U ) – time ( t ) curve will be maximum when t = T β . The value of β is ______.

Step-by-step solution

Given here, displacement x = A   sin   ω t Now, potential energy of a simple harmonic oscillator is U = 1 2 m ω 2 x 2 = 1 2 m ω 2 A 2 sin 2 ω t , So, slope of potential energy and time curve is d U d t = 1 2 m ω 3 A 2 2 sin 2 ω t Now, d U d t m a x will be maximum when sin 2 ω t = 1 ⇒ 2 ω t = π 2 ⇒ 2 2 π T t = π 2 ⇒ t = T 8 ⇒ β = 8